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Real birational implicitization for statistical models

Tobias Boege, Liam Solus

Abstract

We derive an implicit description of the image of a semialgebraic set under a birational map, provided that the denominators of the map are positive on the set. For statistical models which are globally rationally identifiable, this yields model-defining constraints which facilitate model membership testing, representation learning, and model equivalence tests. Many examples illustrate the applicability of our results. The implicit equations recover well-known Markov properties of classical graphical models, as well as other well-studied equations such as the Verma constraint. They also provide Markov properties for generalizations of these frameworks, such as colored or interventional graphical models, staged trees, and the recently introduced Lyapunov models. Under a further mild assumption, we show that our implicit equations generate the vanishing ideal of the model up to a saturation, generalizing previous results of Geiger, Meek and Sturmfels, Duarte and Görgen, Sullivant, and others.

Real birational implicitization for statistical models

Abstract

We derive an implicit description of the image of a semialgebraic set under a birational map, provided that the denominators of the map are positive on the set. For statistical models which are globally rationally identifiable, this yields model-defining constraints which facilitate model membership testing, representation learning, and model equivalence tests. Many examples illustrate the applicability of our results. The implicit equations recover well-known Markov properties of classical graphical models, as well as other well-studied equations such as the Verma constraint. They also provide Markov properties for generalizations of these frameworks, such as colored or interventional graphical models, staged trees, and the recently introduced Lyapunov models. Under a further mild assumption, we show that our implicit equations generate the vanishing ideal of the model up to a saturation, generalizing previous results of Geiger, Meek and Sturmfels, Duarte and Görgen, Sullivant, and others.

Paper Structure

This paper contains 19 sections, 13 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real algebra
  4. Localization and saturation
  5. The Positivstellensatz
  6. Birational implicitization
  7. Full birational isomorphisms
  8. Transferring ideals, cones and monoids
  9. Birational implicitization
  10. Ambirational statistical models
  11. Linear concentration models
  12. Linear structural equation models
  13. Interventions and colored DAGs
  14. Staged tree models
  15. Lyapunov models
  16. ...and 4 more sections

Key Result

lemma 2.1

Let $A$ be an integral domain, $S \subseteq A$ a monoid and $\mathscr{I} \subseteq A$ an ideal.

Figures (5)

  • Figure 1: A linear and a polyhedral concentration model.
  • Figure 2: Semialgebraic submodels of the colored DAG model from \ref{['ex:ColoredDAG']}.
  • Figure 3: Three staged trees representing models on four jointly distributed binary variables $X_1,X_2,X_3,X_4$. Two first two trees are model equivalent, while the third one is not.
  • Figure 4: Submodels of the complete DAG Lyapunov model on $3$ vertices.
  • Figure 5: Elimination vs. saturation for computing the vanishing ideal of a Gaussian DAG model.

Theorems & Definitions (42)

  • lemma 2.1
  • lemma 2.2
  • proof
  • lemma 2.3
  • theorem 2.4: Formal Positivstellensatz RealAlgebra
  • corollary 2.5: Positivstellensatz
  • definition 3.1
  • lemma 3.2
  • proof
  • lemma 3.3
  • ...and 32 more